Decidability of string graphs

Abstract

We show that string graphs can be recognized in nondeterministic exponential time by giving an exponential upper bound on the number of intersections for a drawing realizing the string graph in the plane. This upper bound confirms a conjecture by Kratochv\'{\i}l and Matou\v{s}ek~\cite{KM91} and settles the long-standing open problem of the decidability of string graph recognition (Sinden~\cite{S66}, Graham~\cite{G76}). Finally we show how to apply the result to solve another old open problem: deciding the existence of Euler diagrams, a central problem of topological inference (Grigni, Papadias, Papadimitriou~\cite{GPP95}).